Elements of Morse-Bott theory for elliptic differential equations
We present the compactness theorem for the harmonic map equation perturbed by a quasilinear term. It implies that under some conditions the number of solutions of the generically perturbed equation is finite. We describe how one can count those independently on a perturbation and compute this number in Morse-Bott cases. We outline the general framework for this (discussing index for solutions and computing first Stiefel-Whitney class for the moduli space) and mention other examples.